Preheat oven to 375°F. Spray 8x8 pan with cooking spray. Break the eggs into a small bowl and beat.
Cut each donut into 8 pieces and layer then in the bottom of the prepared dish. In a heavy, large saucepan, heat the sugar over medium high heat. Do not stir. Bring to a boil and caramelize the sugar. As the sugar begins to melt, swirl the pan occasionally to ensure even melting and caramelization. Continue to boil until the all the sugar has melted and the color is deep amber.
Take the pan off the heat and whisk in 1 cup of the milk. Be careful, the mixture will spatter and spurt. Return the pan to low heat and whisk in the rest of the milk. If the sugar seizes up into toffee-like chunks, just keep whisking the mixture over low heat until they all dissolve again. Remove from heat.
Pour about ¼ cup of the sugar/coconut mixture into the eggs and whisk to combine then pour the egg mixture back onto the pan and whisk to combine. Whisk in the vanilla. Pour the mixture over the cut up donuts and bake for 20 minutes. Let cool slightly and serve warm.